import{_ as n,c as a,a as l,b as Q,d as m,o as t}from"./app-DazB1p5v.js";const s={},e={class:"MathJax",jax:"SVG",style:{position:"relative"}},d={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.138ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 503 636","aria-hidden":"true"},r={class:"MathJax",jax:"SVG",style:{position:"relative"}},o={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.041ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 460 636","aria-hidden":"true"},H={class:"MathJax",jax:"SVG",style:{position:"relative"}},i={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"28.535ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 12612.7 1000","aria-hidden":"true"},h={class:"MathJax",jax:"SVG",style:{position:"relative"}},g={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.054ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 466 453","aria-hidden":"true"},p={class:"MathJax",jax:"SVG",style:{position:"relative"}},L={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"29.52ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13048 1000","aria-hidden":"true"},M={class:"MathJax",jax:"SVG",style:{position:"relative"}},f={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.054ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 466 453","aria-hidden":"true"},V={class:"MathJax",jax:"SVG",style:{position:"relative"}},u={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"1.593ex",role:"img",focusable:"false",viewBox:"0 -694 520 704","aria-hidden":"true"},Z={class:"MathJax",jax:"SVG",style:{position:"relative"}},c={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.014ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 8404.2 1000","aria-hidden":"true"},w={class:"MathJax",jax:"SVG",style:{position:"relative"}},x={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.178ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2288.7 1000","aria-hidden":"true"},v={class:"MathJax",jax:"SVG",style:{position:"relative"}},y={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"5.3ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2342.7 1000","aria-hidden":"true"},D={class:"MathJax",jax:"SVG",style:{position:"relative"}},j={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.136ex",height:"1.984ex",role:"img",focusable:"false",viewBox:"0 -683 4480.2 877","aria-hidden":"true"},b={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},k={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.838ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 12304.3 1000","aria-hidden":"true"},C={start:"2"},B={class:"MathJax",jax:"SVG",style:{position:"relative"}},S={style:{"vertical-align":"-0.05ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.719ex",height:"1.645ex",role:"img",focusable:"false",viewBox:"0 -705 760 727","aria-hidden":"true"},J={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},G={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.407ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 12113.7 1153.7","aria-hidden":"true"},R={class:"MathJax",jax:"SVG",style:{position:"relative"}},O={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"13.744ex",height:"1.946ex",role:"img",focusable:"false",viewBox:"0 -666 6074.8 860","aria-hidden":"true"},E={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},A={style:{"vertical-align":"-2.036ex"},xmlns:"http://www.w3.org/2000/svg",width:"39.957ex",height:"5.204ex",role:"img",focusable:"false",viewBox:"0 -1400 17661 2300","aria-hidden":"true"},F={start:"2"},q={class:"MathJax",jax:"SVG",style:{position:"relative"}},X={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.334ex",height:"1.715ex",role:"img",focusable:"false",viewBox:"0 -676 2799.6 758","aria-hidden":"true"},N={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},P={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"24.305ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 10742.7 1000","aria-hidden":"true"},z={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},I={style:{"vertical-align":"-0.464ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.986ex",height:"2.034ex",role:"img",focusable:"false",viewBox:"0 -694 7066 899","aria-hidden":"true"},K={class:"MathJax",jax:"SVG",style:{position:"relative"}},U={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"8.718ex",height:"1.756ex",role:"img",focusable:"false",viewBox:"0 -694 3853.6 776","aria-hidden":"true"},W={start:"3"},Y={class:"MathJax",jax:"SVG",style:{position:"relative"}},$={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.554ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4222.7 1000","aria-hidden":"true"},_={class:"MathJax",jax:"SVG",style:{position:"relative"}},Q1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"11.816ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 5222.7 1000","aria-hidden":"true"},T1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},a1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"34.312ex",height:"2.581ex",role:"img",focusable:"false",viewBox:"0 -891 15165.9 1141","aria-hidden":"true"},t1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},m1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"38.319ex",height:"2.581ex",role:"img",focusable:"false",viewBox:"0 -891 16937 1141","aria-hidden":"true"},l1={class:"MathJax",jax:"SVG",style:{position:"relative"}},n1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.422ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 9026.5 1000","aria-hidden":"true"},s1={class:"MathJax",jax:"SVG",style:{position:"relative"}},e1={style:{"vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"13.472ex",height:"1.781ex",role:"img",focusable:"false",viewBox:"0 -705 5954.6 787","aria-hidden":"true"},d1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},r1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.037ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 8414.2 1153.7","aria-hidden":"true"},o1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},H1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.694ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 12240.9 1153.7","aria-hidden":"true"},i1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},h1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.445ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 9036.5 1153.7","aria-hidden":"true"},g1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},p1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.272ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 8960.2 1000","aria-hidden":"true"},L1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},M1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"14.914ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 6592 1000","aria-hidden":"true"},f1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},V1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"24.991ex",height:"2.7ex",role:"img",focusable:"false",viewBox:"0 -943.3 11046.1 1193.3","aria-hidden":"true"},u1={id:"_5-1-与互质",tabindex:"-1"},Z1={class:"header-anchor",href:"#_5-1-与互质"},c1={class:"MathJax",jax:"SVG",style:{position:"relative"}},w1={style:{"vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},x1={class:"MathJax",jax:"SVG",style:{position:"relative"}},v1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 600 453","aria-hidden":"true"},y1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},D1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.779ex",height:"2.7ex",role:"img",focusable:"false",viewBox:"0 -943.3 9184.1 1193.3","aria-hidden":"true"},j1={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},b1={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"30.038ex",height:"3.193ex",role:"img",focusable:"false",viewBox:"0 -1161.1 13276.8 1411.1","aria-hidden":"true"},k1={id:"_5-2-与不是互质关系",tabindex:"-1"},C1={class:"header-anchor",href:"#_5-2-与不是互质关系"},B1={class:"MathJax",jax:"SVG",style:{position:"relative"}},S1={style:{"vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},J1={class:"MathJax",jax:"SVG",style:{position:"relative"}},G1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 600 453","aria-hidden":"true"},R1={class:"MathJax",jax:"SVG",style:{position:"relative"}},O1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.357ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 600 453","aria-hidden":"true"},E1={class:"MathJax",jax:"SVG",style:{position:"relative"}},A1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.138ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 503 636","aria-hidden":"true"},F1={class:"MathJax",jax:"SVG",style:{position:"relative"}},q1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.041ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 460 636","aria-hidden":"true"},X1={class:"MathJax",jax:"SVG",style:{position:"relative"}},N1={style:{"vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"2.378ex",height:"1.545ex",role:"img",focusable:"false",viewBox:"0 -683 1051 683","aria-hidden":"true"},P1={class:"MathJax",jax:"SVG",style:{position:"relative"}},z1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.317ex",height:"2.009ex",role:"img",focusable:"false",viewBox:"0 -694 1024 888","aria-hidden":"true"},I1={class:"MathJax",jax:"SVG",style:{position:"relative"}},K1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"2.219ex",height:"2.009ex",role:"img",focusable:"false",viewBox:"0 -694 981 888","aria-hidden":"true"},U1={class:"MathJax",jax:"SVG",style:{position:"relative"}},W1={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.712ex",height:"2.009ex",role:"img",focusable:"false",viewBox:"0 -694 3408.6 888","aria-hidden":"true"},Y1={class:"MathJax",jax:"SVG",style:{position:"relative"}},$1={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.179ex",height:"1.595ex",role:"img",focusable:"false",viewBox:"0 -694 521 705","aria-hidden":"true"},_1={class:"MathJax",jax:"SVG",style:{position:"relative"}},Q2={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.041ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 460 636","aria-hidden":"true"},T2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},a2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"21.661ex",height:"2.71ex",role:"img",focusable:"false",viewBox:"0 -948 9574.2 1198","aria-hidden":"true"},t2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},m2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"34.892ex",height:"2.7ex",role:"img",focusable:"false",viewBox:"0 -943.3 15422.4 1193.3","aria-hidden":"true"},l2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},n2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"21.644ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 9566.4 1153.7","aria-hidden":"true"},s2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},e2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.799ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 6983.2 1153.7","aria-hidden":"true"},d2={class:"MathJax",jax:"SVG",style:{position:"relative"}},r2={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.817ex",height:"1.441ex",role:"img",focusable:"false",viewBox:"0 -626 361 637","aria-hidden":"true"},o2={class:"MathJax",jax:"SVG",style:{position:"relative"}},H2={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.138ex",height:"1.439ex",role:"img",focusable:"false",viewBox:"0 -442 503 636","aria-hidden":"true"},i2={class:"MathJax",jax:"SVG",style:{position:"relative"}},h2={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.416ex",height:"2.156ex",role:"img",focusable:"false",viewBox:"0 -759 2836 953","aria-hidden":"true"},g2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},p2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"17.565ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 7763.7 1153.7","aria-hidden":"true"},L2={class:"MathJax",jax:"SVG",style:{position:"relative"}},M2={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"14.88ex",height:"2.009ex",role:"img",focusable:"false",viewBox:"0 -694 6576.8 888","aria-hidden":"true"},f2={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},V2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.445ex",height:"2.61ex",role:"img",focusable:"false",viewBox:"0 -903.7 9036.5 1153.7","aria-hidden":"true"},u2={class:"MathJax",jax:"SVG",style:{position:"relative"}},Z2={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.553ex",height:"1.758ex",role:"img",focusable:"false",viewBox:"0 -583 2896.6 777","aria-hidden":"true"},c2={class:"MathJax",jax:"SVG",style:{position:"relative"}},w2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.863ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3475.3 1000","aria-hidden":"true"},x2={class:"MathJax",jax:"SVG",style:{position:"relative"}},v2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.014ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 8404.2 1000","aria-hidden":"true"},y2={class:"MathJax",jax:"SVG",style:{position:"relative"}},D2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"15.829ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 6996.2 1000","aria-hidden":"true"},j2={class:"MathJax",jax:"SVG",style:{position:"relative"}},b2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.686ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 5607.1 1000","aria-hidden":"true"},k2={class:"MathJax",jax:"SVG",style:{position:"relative"}},C2={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.054ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 466 453","aria-hidden":"true"},B2={class:"MathJax",jax:"SVG",style:{position:"relative"}},S2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.466ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 1974 1000","aria-hidden":"true"},J2={class:"MathJax",jax:"SVG",style:{position:"relative"}},G2={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.054ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 466 453","aria-hidden":"true"},R2={class:"MathJax",jax:"SVG",style:{position:"relative"}},O2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.575ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3348 1000","aria-hidden":"true"},E2={class:"MathJax",jax:"SVG",style:{position:"relative"}},A2={style:{"vertical-align":"-0.023ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.176ex",height:"1.593ex",role:"img",focusable:"false",viewBox:"0 -694 520 704","aria-hidden":"true"},F2={class:"MathJax",jax:"SVG",style:{position:"relative"}},q2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.014ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 8404.2 1000","aria-hidden":"true"},X2={class:"MathJax",jax:"SVG",style:{position:"relative"}},N2={style:{"vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.054ex",height:"1.025ex",role:"img",focusable:"false",viewBox:"0 -442 466 453","aria-hidden":"true"},P2={class:"MathJax",jax:"SVG",style:{position:"relative"}},z2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"19.132ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 8456.2 1000","aria-hidden":"true"},I2={class:"MathJax",jax:"SVG",style:{position:"relative"}},K2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.926ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 5713.1 1000","aria-hidden":"true"},U2={class:"MathJax",jax:"SVG",style:{position:"relative"}},W2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.863ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3475.3 1000","aria-hidden":"true"},Y2={class:"MathJax",jax:"SVG",style:{position:"relative"}},$2={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.575ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3348 1000","aria-hidden":"true"},_2={class:"MathJax",jax:"SVG",style:{position:"relative"}},Q3={style:{"vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"13.744ex",height:"1.968ex",role:"img",focusable:"false",viewBox:"0 -676 6074.8 870","aria-hidden":"true"},T3={class:"MathJax",jax:"SVG",style:{position:"relative"}},a3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"23.807ex",height:"2.452ex",role:"img",focusable:"false",viewBox:"0 -833.9 10522.6 1083.9","aria-hidden":"true"},t3={class:"MathJax",jax:"SVG",style:{position:"relative"}},m3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"37.137ex",height:"2.452ex",role:"img",focusable:"false",viewBox:"0 -833.9 16414.7 1083.9","aria-hidden":"true"},l3={class:"MathJax",jax:"SVG",style:{position:"relative"}},n3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.838ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4348.4 1000","aria-hidden":"true"},s3={class:"MathJax",jax:"SVG",style:{position:"relative"}},e3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"9.838ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4348.4 1000","aria-hidden":"true"},d3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},r3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"29.471ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 13026 1000","aria-hidden":"true"},o3={class:"MathJax",jax:"SVG",display:"true",style:{position:"relative"}},H3={style:{"vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"27.072ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 11965.6 1000","aria-hidden":"true"};function i3(h3,T){return t(),a("div",null,[T[231]||(T[231]=l('
1977年,三位数学家RonRivest、Adi Shamir 和 Leonard Adleman 设计了一种算法,可以实现非对称加密。这种算法用他们三个人的名字命名,叫做RSA算法。
',5)),Q("ol",null,[Q("li",null,[T[4]||(T[4]=m("随机选择两个不相等的质数")),Q("mjx-container",e,[(t(),a("svg",d,T[0]||(T[0]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"})])])],-1)]))),T[1]||(T[1]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p")])],-1))]),T[5]||(T[5]=m("和")),Q("mjx-container",r,[(t(),a("svg",o,T[2]||(T[2]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45E",d:"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"})])])],-1)]))),T[3]||(T[3]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"q")])],-1))])]),Q("li",null,[T[8]||(T[8]=m("计算")),Q("mjx-container",H,[(t(),a("svg",i,T[6]||(T[6]=[l('',1)]))),T[7]||(T[7]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"n"),Q("mo",null,"="),Q("mi",null,"q"),Q("mi",null,"p"),Q("mo",null,","),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mi",null,"p"),Q("mo",null,"−"),Q("mn",null,"1"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"q"),Q("mo",null,"−"),Q("mn",null,"1"),Q("mo",{stretchy:"false"},")")])],-1))])]),Q("li",null,[T[13]||(T[13]=m("选择一个整数")),Q("mjx-container",h,[(t(),a("svg",g,T[9]||(T[9]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D452",d:"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"})])])],-1)]))),T[10]||(T[10]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e")])],-1))]),T[14]||(T[14]=m(",满足")),Q("mjx-container",p,[(t(),a("svg",L,T[11]||(T[11]=[l('',1)]))),T[12]||(T[12]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"1"),Q("mo",null,"<"),Q("mi",null,"e"),Q("mo",null,"<"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,","),Q("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"e"),Q("mo",null,","),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"1")])],-1))])]),Q("li",null,[T[21]||(T[21]=m("计算")),Q("mjx-container",M,[(t(),a("svg",f,T[15]||(T[15]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D452",d:"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"})])])],-1)]))),T[16]||(T[16]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e")])],-1))]),T[22]||(T[22]=m("的模反元素")),Q("mjx-container",V,[(t(),a("svg",u,T[17]||(T[17]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D451",d:"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"})])])],-1)]))),T[18]||(T[18]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"d")])],-1))]),T[23]||(T[23]=m(",满足")),Q("mjx-container",Z,[(t(),a("svg",c,T[19]||(T[19]=[l('',1)]))),T[20]||(T[20]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e"),Q("mi",null,"d"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"0.444em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))])]),Q("li",null,[T[28]||(T[28]=m("公钥为")),Q("mjx-container",w,[(t(),a("svg",x,T[24]||(T[24]=[l('',1)]))),T[25]||(T[25]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"e"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[29]||(T[29]=m(",密钥为")),Q("mjx-container",v,[(t(),a("svg",y,T[26]||(T[26]=[l('',1)]))),T[27]||(T[27]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"d"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))])])]),T[232]||(T[232]=Q("h2",{id:"_3-加密和解密",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_3-加密和解密"},[Q("span",null,"3. 加密和解密")])],-1)),Q("ol",null,[Q("li",null,[T[32]||(T[32]=m("加密明文")),Q("mjx-container",D,[(t(),a("svg",j,T[30]||(T[30]=[l('',1)]))),T[31]||(T[31]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"M"),Q("mo",null,","),Q("mi",null,"M"),Q("mo",null,"<"),Q("mi",null,"n")])],-1))]),T[33]||(T[33]=m(","))])]),Q("mjx-container",b,[(t(),a("svg",k,T[34]||(T[34]=[l('',1)]))),T[35]||(T[35]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"C"),Q("mo",null,"="),Q("mi",null,"E"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"M"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msup",null,[Q("mi",null,"M"),Q("mi",null,"e")]),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),Q("ol",C,[Q("li",null,[T[38]||(T[38]=m("解密密文")),Q("mjx-container",B,[(t(),a("svg",S,T[36]||(T[36]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D436",d:"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"})])])],-1)]))),T[37]||(T[37]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"C")])],-1))]),T[39]||(T[39]=m(","))])]),Q("mjx-container",J,[(t(),a("svg",G,T[40]||(T[40]=[l('',1)]))),T[41]||(T[41]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"M"),Q("mo",null,"="),Q("mi",null,"D"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"C"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msup",null,[Q("mi",null,"C"),Q("mi",null,"d")]),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[233]||(T[233]=Q("h2",{id:"_4-举例",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_4-举例"},[Q("span",null,"4. 举例")])],-1)),Q("ol",null,[Q("li",null,[T[44]||(T[44]=m("爱丽丝选择了")),Q("mjx-container",R,[(t(),a("svg",O,T[42]||(T[42]=[l('',1)]))),T[43]||(T[43]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p"),Q("mo",null,"="),Q("mn",null,"61"),Q("mo",null,","),Q("mi",null,"q"),Q("mo",null,"="),Q("mn",null,"53")])],-1))]),T[45]||(T[45]=m(", 计算"))])]),Q("mjx-container",E,[(t(),a("svg",A,T[46]||(T[46]=[l('',1)]))),T[47]||(T[47]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[Q("mtr",null,[Q("mtd",null,[Q("mi",null,"n")]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mi",null,"p"),Q("mi",null,"q"),Q("mo",null,"="),Q("mn",null,"61"),Q("mo",null,"×"),Q("mn",null,"53"),Q("mo",null,"="),Q("mn",null,"3233")])]),Q("mtr",null,[Q("mtd",null,[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")]),Q("mtd",null,[Q("mi"),Q("mo",null,"="),Q("mo",{stretchy:"false"},"("),Q("mi",null,"p"),Q("mo",null,"−"),Q("mn",null,"1"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"q"),Q("mo",null,"−"),Q("mn",null,"1"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"360"),Q("mo",null,"×"),Q("mn",null,"52"),Q("mo",null,"="),Q("mn",null,"3120")])])])])],-1))]),Q("ol",F,[Q("li",null,[T[50]||(T[50]=m("爱丽丝选择")),Q("mjx-container",q,[(t(),a("svg",X,T[48]||(T[48]=[l('',1)]))),T[49]||(T[49]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e"),Q("mo",null,"="),Q("mn",null,"17")])],-1))]),T[51]||(T[51]=m(",解同余方程"))])]),Q("mjx-container",N,[(t(),a("svg",P,T[52]||(T[52]=[l('',1)]))),T[53]||(T[53]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mn",null,"17"),Q("mo",null,"×"),Q("mi",null,"d"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mn",null,"3120"),Q("mo",{stretchy:"false"},")")])],-1))]),T[234]||(T[234]=Q("p",null,"使用扩展欧几里得算法解方程",-1)),Q("mjx-container",z,[(t(),a("svg",I,T[54]||(T[54]=[l('',1)]))),T[55]||(T[55]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mn",null,"17"),Q("mi",null,"d"),Q("mo",null,"+"),Q("mn",null,"3120"),Q("mi",null,"y"),Q("mo",null,"="),Q("mn",null,"1")])],-1))]),Q("p",null,[T[58]||(T[58]=m("得到")),Q("mjx-container",K,[(t(),a("svg",U,T[56]||(T[56]=[l('',1)]))),T[57]||(T[57]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"d"),Q("mo",null,"="),Q("mn",null,"2753")])],-1))])]),Q("ol",W,[Q("li",null,[Q("p",null,[T[63]||(T[63]=m("公钥为")),Q("mjx-container",Y,[(t(),a("svg",$,T[59]||(T[59]=[l('',1)]))),T[60]||(T[60]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mn",null,"17"),Q("mo",null,","),Q("mn",null,"3233"),Q("mo",{stretchy:"false"},")")])],-1))]),T[64]||(T[64]=m(", 密钥为")),Q("mjx-container",_,[(t(),a("svg",Q1,T[61]||(T[61]=[l('',1)]))),T[62]||(T[62]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{stretchy:"false"},"("),Q("mn",null,"2753"),Q("mo",null,","),Q("mn",null,"3233"),Q("mo",{stretchy:"false"},")")])],-1))])])]),T[65]||(T[65]=Q("li",null,[Q("p",null,"鲍勃使用公钥加密明文'A'=65")],-1))]),Q("mjx-container",T1,[(t(),a("svg",a1,T[66]||(T[66]=[l('',1)]))),T[67]||(T[67]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"E"),Q("mo",{stretchy:"false"},"("),Q("mn",null,"65"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msup",null,[Q("mn",null,"65"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"17")])]),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mn",null,"3233"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"2790")])],-1))]),T[235]||(T[235]=Q("ol",{start:"5"},[Q("li",null,"爱丽丝用私钥解密")],-1)),Q("mjx-container",t1,[(t(),a("svg",m1,T[68]||(T[68]=[l('',1)]))),T[69]||(T[69]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"D"),Q("mo",{stretchy:"false"},"("),Q("mn",null,"2790"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msup",null,[Q("mn",null,"2790"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mn",null,"2753")])]),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mn",null,"3233"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"65")])],-1))]),T[236]||(T[236]=Q("h2",{id:"_5-证明",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_5-证明"},[Q("span",null,"5. 证明")])],-1)),Q("p",null,[T[74]||(T[74]=m("根据加密规则")),Q("mjx-container",l1,[(t(),a("svg",n1,T[70]||(T[70]=[l('',1)]))),T[71]||(T[71]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("msup",null,[Q("mi",null,"M"),Q("mi",null,"e")]),Q("mo",null,"≡"),Q("mi",null,"C"),Q("mspace",{width:"0.444em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,","),Q("mi",null,"C")])],-1))]),T[75]||(T[75]=m("可以写成")),Q("mjx-container",s1,[(t(),a("svg",e1,T[72]||(T[72]=[l('',1)]))),T[73]||(T[73]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"C"),Q("mo",null,"="),Q("msup",null,[Q("mi",null,"M"),Q("mi",null,"e")]),Q("mo",null,"−"),Q("mi",null,"k"),Q("mi",null,"n")])],-1))]),T[76]||(T[76]=m(" 下面证明下面的公式成立"))]),Q("mjx-container",d1,[(t(),a("svg",r1,T[77]||(T[77]=[l('',1)]))),T[78]||(T[78]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mi",null,"C"),Q("mi",null,"d")]),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[237]||(T[237]=Q("p",null,"将C代入要我们要证明的那个解密规则:",-1)),Q("mjx-container",o1,[(t(),a("svg",H1,T[79]||(T[79]=[l('',1)]))),T[80]||(T[80]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("msup",null,[Q("mi",null,"M"),Q("mi",null,"e")]),Q("mo",null,"−"),Q("mi",null,"k"),Q("mi",null,"n"),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"d")])]),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[238]||(T[238]=Q("p",null,"它等同于求证",-1)),Q("mjx-container",i1,[(t(),a("svg",h1,T[81]||(T[81]=[l('',1)]))),T[82]||(T[82]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"e"),Q("mi",null,"d")])]),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[239]||(T[239]=Q("p",null,"由于",-1)),Q("mjx-container",g1,[(t(),a("svg",p1,T[83]||(T[83]=[l('',1)]))),T[84]||(T[84]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"e"),Q("mi",null,"d"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[240]||(T[240]=Q("p",null,"所以",-1)),Q("mjx-container",L1,[(t(),a("svg",M1,T[85]||(T[85]=[l('',1)]))),T[86]||(T[86]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mi",null,"e"),Q("mi",null,"d"),Q("mo",null,"="),Q("mi",null,"h"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mn",null,"1")])],-1))]),T[241]||(T[241]=Q("p",null,"将ed代入得到:",-1)),Q("mjx-container",f1,[(t(),a("svg",V1,T[87]||(T[87]=[l('',1)]))),T[88]||(T[88]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"h"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"+"),Q("mn",null,"1")])]),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[242]||(T[242]=Q("p",null,"接下来,分成两种情况证明上面这个式子。",-1)),Q("h4",u1,[Q("a",Z1,[Q("span",null,[T[93]||(T[93]=m("5.1 ")),Q("mjx-container",c1,[(t(),a("svg",w1,T[89]||(T[89]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"})])])],-1)]))),T[90]||(T[90]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"M")])],-1))]),T[94]||(T[94]=m("与")),Q("mjx-container",x1,[(t(),a("svg",v1,T[91]||(T[91]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45B",d:"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"})])])],-1)]))),T[92]||(T[92]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"n")])],-1))]),T[95]||(T[95]=m("互质"))])])]),T[243]||(T[243]=Q("p",null,"根据欧拉定理,此时",-1)),Q("mjx-container",y1,[(t(),a("svg",D1,T[96]||(T[96]=[l('',1)]))),T[97]||(T[97]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])]),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[244]||(T[244]=Q("p",null,"得到",-1)),Q("mjx-container",j1,[(t(),a("svg",b1,T[98]||(T[98]=[l('',1)]))),T[99]||(T[99]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{stretchy:"false"},"("),Q("msup",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])]),Q("mo",{stretchy:"false"},")")]),Q("mi",null,"h")]),Q("mo",null,"×"),Q("mi",null,"M"),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[245]||(T[245]=Q("p",null,"原式得到证明。",-1)),Q("h4",k1,[Q("a",C1,[Q("span",null,[T[104]||(T[104]=m("5.2 ")),Q("mjx-container",B1,[(t(),a("svg",S1,T[100]||(T[100]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"})])])],-1)]))),T[101]||(T[101]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"M")])],-1))]),T[105]||(T[105]=m("与")),Q("mjx-container",J1,[(t(),a("svg",G1,T[102]||(T[102]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45B",d:"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"})])])],-1)]))),T[103]||(T[103]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"n")])],-1))]),T[106]||(T[106]=m("不是互质关系"))])])]),Q("p",null,[T[125]||(T[125]=m("此时,由于")),Q("mjx-container",R1,[(t(),a("svg",O1,T[107]||(T[107]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45B",d:"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z"})])])],-1)]))),T[108]||(T[108]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"n")])],-1))]),T[126]||(T[126]=m("等于质数")),Q("mjx-container",E1,[(t(),a("svg",A1,T[109]||(T[109]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"})])])],-1)]))),T[110]||(T[110]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p")])],-1))]),T[127]||(T[127]=m("和")),Q("mjx-container",F1,[(t(),a("svg",q1,T[111]||(T[111]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45E",d:"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"})])])],-1)]))),T[112]||(T[112]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"q")])],-1))]),T[128]||(T[128]=m("的乘积,所以")),Q("mjx-container",X1,[(t(),a("svg",N1,T[113]||(T[113]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D440",d:"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z"})])])],-1)]))),T[114]||(T[114]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"M")])],-1))]),T[129]||(T[129]=m("必然等于")),Q("mjx-container",P1,[(t(),a("svg",z1,T[115]||(T[115]=[l('',1)]))),T[116]||(T[116]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"k"),Q("mi",null,"p")])],-1))]),T[130]||(T[130]=m("或")),Q("mjx-container",I1,[(t(),a("svg",K1,T[117]||(T[117]=[l('',1)]))),T[118]||(T[118]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"k"),Q("mi",null,"q")])],-1))]),T[131]||(T[131]=m("。 以")),Q("mjx-container",U1,[(t(),a("svg",W1,T[119]||(T[119]=[l('',1)]))),T[120]||(T[120]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"M"),Q("mo",null,"="),Q("mi",null,"k"),Q("mi",null,"p")])],-1))]),T[132]||(T[132]=m("为例,考虑到这时")),Q("mjx-container",Y1,[(t(),a("svg",$1,T[121]||(T[121]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D458",d:"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z"})])])],-1)]))),T[122]||(T[122]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"k")])],-1))]),T[133]||(T[133]=m("与")),Q("mjx-container",_1,[(t(),a("svg",Q2,T[123]||(T[123]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45E",d:"M33 157Q33 258 109 349T280 441Q340 441 372 389Q373 390 377 395T388 406T404 418Q438 442 450 442Q454 442 457 439T460 434Q460 425 391 149Q320 -135 320 -139Q320 -147 365 -148H390Q396 -156 396 -157T393 -175Q389 -188 383 -194H370Q339 -192 262 -192Q234 -192 211 -192T174 -192T157 -193Q143 -193 143 -185Q143 -182 145 -170Q149 -154 152 -151T172 -148Q220 -148 230 -141Q238 -136 258 -53T279 32Q279 33 272 29Q224 -10 172 -10Q117 -10 75 30T33 157ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"})])])],-1)]))),T[124]||(T[124]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"q")])],-1))]),T[134]||(T[134]=m("必然互质,则根据欧拉定理,下面的式子成立:"))]),Q("mjx-container",T2,[(t(),a("svg",a2,T[135]||(T[135]=[l('',1)]))),T[136]||(T[136]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mi",null,"p"),Q("mo",{stretchy:"false"},")")]),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"q"),Q("mo",null,"−"),Q("mn",null,"1")])]),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"q"),Q("mo",{stretchy:"false"},")")])],-1))]),T[246]||(T[246]=Q("p",null,"进一步得到",-1)),Q("mjx-container",t2,[(t(),a("svg",m2,T[137]||(T[137]=[l('',1)]))),T[138]||(T[138]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mi",null,"p"),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"q"),Q("mo",null,"−"),Q("mn",null,"1")])]),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"h"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"p"),Q("mo",null,"−"),Q("mn",null,"1"),Q("mo",{stretchy:"false"},")")])]),Q("mo",null,"×"),Q("mi",null,"k"),Q("mi",null,"p"),Q("mo",null,"≡"),Q("mi",null,"k"),Q("mi",null,"p"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"q"),Q("mo",{stretchy:"false"},")")])],-1))]),T[247]||(T[247]=Q("p",null,"即",-1)),Q("mjx-container",l2,[(t(),a("svg",n2,T[139]||(T[139]=[l('',1)]))),T[140]||(T[140]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mi",null,"p"),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"e"),Q("mi",null,"d")])]),Q("mo",null,"≡"),Q("mi",null,"k"),Q("mi",null,"p"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"q"),Q("mo",{stretchy:"false"},")")])],-1))]),T[248]||(T[248]=Q("p",null,"将它改写成下面的等式",-1)),Q("mjx-container",s2,[(t(),a("svg",e2,T[141]||(T[141]=[l('',1)]))),T[142]||(T[142]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mi",null,"p"),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"e"),Q("mi",null,"d")])]),Q("mo",null,"="),Q("mi",null,"t"),Q("mi",null,"q"),Q("mo",null,"+"),Q("mi",null,"k"),Q("mi",null,"p")])],-1))]),Q("p",null,[T[149]||(T[149]=m("这时")),Q("mjx-container",d2,[(t(),a("svg",r2,T[143]||(T[143]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D461",d:"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"})])])],-1)]))),T[144]||(T[144]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"t")])],-1))]),T[150]||(T[150]=m("必然能被")),Q("mjx-container",o2,[(t(),a("svg",H2,T[145]||(T[145]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D45D",d:"M23 287Q24 290 25 295T30 317T40 348T55 381T75 411T101 433T134 442Q209 442 230 378L240 387Q302 442 358 442Q423 442 460 395T497 281Q497 173 421 82T249 -10Q227 -10 210 -4Q199 1 187 11T168 28L161 36Q160 35 139 -51T118 -138Q118 -144 126 -145T163 -148H188Q194 -155 194 -157T191 -175Q188 -187 185 -190T172 -194Q170 -194 161 -194T127 -193T65 -192Q-5 -192 -24 -194H-32Q-39 -187 -39 -183Q-37 -156 -26 -148H-6Q28 -147 33 -136Q36 -130 94 103T155 350Q156 355 156 364Q156 405 131 405Q109 405 94 377T71 316T59 280Q57 278 43 278H29Q23 284 23 287ZM178 102Q200 26 252 26Q282 26 310 49T356 107Q374 141 392 215T411 325V331Q411 405 350 405Q339 405 328 402T306 393T286 380T269 365T254 350T243 336T235 326L232 322Q232 321 229 308T218 264T204 212Q178 106 178 102Z"})])])],-1)]))),T[146]||(T[146]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p")])],-1))]),T[151]||(T[151]=m("整除,即 ")),Q("mjx-container",i2,[(t(),a("svg",h2,T[147]||(T[147]=[l('',1)]))),T[148]||(T[148]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"t"),Q("mo",null,"="),Q("msup",null,[Q("mi",null,"t"),Q("mo",{"data-mjx-alternate":"1"},"′")]),Q("mi",null,"p")])],-1))])]),Q("mjx-container",g2,[(t(),a("svg",p2,T[152]||(T[152]=[l('',1)]))),T[153]||(T[153]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("mo",{stretchy:"false"},"("),Q("mi",null,"k"),Q("mi",null,"p"),Q("msup",null,[Q("mo",{stretchy:"false"},")"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"e"),Q("mi",null,"d")])]),Q("mo",null,"="),Q("msup",null,[Q("mi",null,"t"),Q("mo",{"data-mjx-alternate":"1"},"′")]),Q("mi",null,"p"),Q("mi",null,"q"),Q("mo",null,"+"),Q("mi",null,"k"),Q("mi",null,"p")])],-1))]),Q("p",null,[T[156]||(T[156]=m("因为 ")),Q("mjx-container",L2,[(t(),a("svg",M2,T[154]||(T[154]=[l('',1)]))),T[155]||(T[155]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"m"),Q("mo",null,"="),Q("mi",null,"k"),Q("mi",null,"p"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",null,"="),Q("mi",null,"p"),Q("mi",null,"q")])],-1))]),T[157]||(T[157]=m(",所以"))]),Q("mjx-container",f2,[(t(),a("svg",V2,T[158]||(T[158]=[l('',1)]))),T[159]||(T[159]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msup",null,[Q("mi",null,"M"),Q("mrow",{"data-mjx-texclass":"ORD"},[Q("mi",null,"e"),Q("mi",null,"d")])]),Q("mo",null,"≡"),Q("mi",null,"M"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[249]||(T[249]=Q("p",null,"原式得到证明。",-1)),T[250]||(T[250]=Q("h2",{id:"_6-密钥对的个数",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_6-密钥对的个数"},[Q("span",null,"6. 密钥对的个数")])],-1)),Q("p",null,[T[164]||(T[164]=m("对于选定的")),Q("mjx-container",u2,[(t(),a("svg",Z2,T[160]||(T[160]=[l('',1)]))),T[161]||(T[161]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"n"),Q("mo",null,"="),Q("mi",null,"p"),Q("mi",null,"q")])],-1))]),T[165]||(T[165]=m(",能够产生的密钥对")),Q("mjx-container",c2,[(t(),a("svg",w2,T[162]||(T[162]=[l('',1)]))),T[163]||(T[163]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{fence:"false",stretchy:"false"},"{"),Q("mi",null,"e"),Q("mo",null,","),Q("mi",null,"d"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),T[166]||(T[166]=m("有多少个?"))]),Q("p",null,[T[195]||(T[195]=m("由于")),Q("mjx-container",x2,[(t(),a("svg",v2,T[167]||(T[167]=[l('',1)]))),T[168]||(T[168]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e"),Q("mi",null,"d"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"0.444em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[196]||(T[196]=m(",所以")),Q("mjx-container",y2,[(t(),a("svg",D2,T[169]||(T[169]=[l('',1)]))),T[170]||(T[170]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"e"),Q("mo",null,","),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"1")])],-1))]),T[197]||(T[197]=m(",而且")),Q("mjx-container",j2,[(t(),a("svg",b2,T[171]||(T[171]=[l('',1)]))),T[172]||(T[172]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"1"),Q("mo",null,"<"),Q("mi",null,"e"),Q("mo",null,"<"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[198]||(T[198]=m(",所以")),Q("mjx-container",k2,[(t(),a("svg",C2,T[173]||(T[173]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D452",d:"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"})])])],-1)]))),T[174]||(T[174]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e")])],-1))]),T[199]||(T[199]=m("的个数是和")),Q("mjx-container",B2,[(t(),a("svg",S2,T[175]||(T[175]=[l('',1)]))),T[176]||(T[176]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[200]||(T[200]=m("互质的数的个数,也就是")),Q("mjx-container",J2,[(t(),a("svg",G2,T[177]||(T[177]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D452",d:"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"})])])],-1)]))),T[178]||(T[178]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e")])],-1))]),T[201]||(T[201]=m("有")),Q("mjx-container",R2,[(t(),a("svg",O2,T[179]||(T[179]=[l('',1)]))),T[180]||(T[180]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[202]||(T[202]=m("个 对于")),Q("mjx-container",E2,[(t(),a("svg",A2,T[181]||(T[181]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D451",d:"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z"})])])],-1)]))),T[182]||(T[182]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"d")])],-1))]),T[203]||(T[203]=m(",由于")),Q("mjx-container",F2,[(t(),a("svg",q2,T[183]||(T[183]=[l('',1)]))),T[184]||(T[184]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e"),Q("mi",null,"d"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"0.444em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[204]||(T[204]=m(",当")),Q("mjx-container",X2,[(t(),a("svg",N2,T[185]||(T[185]=[Q("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[Q("g",{"data-mml-node":"math"},[Q("g",{"data-mml-node":"mi"},[Q("path",{"data-c":"1D452",d:"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z"})])])],-1)]))),T[186]||(T[186]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e")])],-1))]),T[205]||(T[205]=m("确定时,方程")),Q("mjx-container",P2,[(t(),a("svg",z2,T[187]||(T[187]=[l('',1)]))),T[188]||(T[188]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"e"),Q("mi",null,"x"),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"0.444em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[206]||(T[206]=m("在")),Q("mjx-container",I2,[(t(),a("svg",K2,T[189]||(T[189]=[l('',1)]))),T[190]||(T[190]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mn",null,"1"),Q("mo",null,"<"),Q("mi",null,"x"),Q("mo",null,"<"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")")])],-1))]),T[207]||(T[207]=m("有且只有一个解,所以密钥对")),Q("mjx-container",U2,[(t(),a("svg",W2,T[191]||(T[191]=[l('',1)]))),T[192]||(T[192]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{fence:"false",stretchy:"false"},"{"),Q("mi",null,"e"),Q("mo",null,","),Q("mi",null,"d"),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),T[208]||(T[208]=m("的个数为")),Q("mjx-container",Y2,[(t(),a("svg",$2,T[193]||(T[193]=[l('',1)]))),T[194]||(T[194]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[209]||(T[209]=m("个"))]),Q("p",null,[T[216]||(T[216]=m("比如")),Q("mjx-container",_2,[(t(),a("svg",Q3,T[210]||(T[210]=[l('',1)]))),T[211]||(T[211]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"p"),Q("mo",null,"="),Q("mn",null,"13"),Q("mo",null,","),Q("mi",null,"q"),Q("mo",null,"="),Q("mn",null,"17")])],-1))]),T[217]||(T[217]=m(",那么")),Q("mjx-container",T3,[(t(),a("svg",a3,T[212]||(T[212]=[l('',1)]))),T[213]||(T[213]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mn",null,"12"),Q("mo",null,"×"),Q("mn",null,"16"),Q("mo",null,"="),Q("msup",null,[Q("mn",null,"2"),Q("mn",null,"6")]),Q("mo",null,"×"),Q("mn",null,"3")])],-1))]),T[218]||(T[218]=m(", ")),Q("mjx-container",t3,[(t(),a("svg",m3,T[214]||(T[214]=[l('',1)]))),T[215]||(T[215]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("msup",null,[Q("mn",null,"2"),Q("mn",null,"6")]),Q("mo",{stretchy:"false"},")"),Q("mo",null,"×"),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mn",null,"3"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msup",null,[Q("mn",null,"2"),Q("mn",null,"5")]),Q("mo",null,"×"),Q("mn",null,"2"),Q("mo",null,"="),Q("mn",null,"64")])],-1))]),T[219]||(T[219]=m(" 所以一共有64对密钥"))]),T[251]||(T[251]=Q("h2",{id:"_7-交换律",tabindex:"-1"},[Q("a",{class:"header-anchor",href:"#_7-交换律"},[Q("span",null,"7. 交换律")])],-1)),Q("p",null,[T[224]||(T[224]=m("当通信的双方使用模数相等的RSA算法时,RSA算法满足交换律,也就是说Alice使用的密钥为")),Q("mjx-container",l3,[(t(),a("svg",n3,T[220]||(T[220]=[l('',1)]))),T[221]||(T[221]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{fence:"false",stretchy:"false"},"{"),Q("msub",null,[Q("mi",null,"e"),Q("mn",null,"1")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"d"),Q("mn",null,"1")]),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),T[225]||(T[225]=m(", Bob使用的密钥对为")),Q("mjx-container",s3,[(t(),a("svg",e3,T[222]||(T[222]=[l('',1)]))),T[223]||(T[223]=Q("mjx-assistive-mml",{unselectable:"on",display:"inline"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[Q("mo",{fence:"false",stretchy:"false"},"{"),Q("msub",null,[Q("mi",null,"e"),Q("mn",null,"2")]),Q("mo",null,","),Q("msub",null,[Q("mi",null,"d"),Q("mn",null,"2")]),Q("mo",null,","),Q("mi",null,"n"),Q("mo",{fence:"false",stretchy:"false"},"}")])],-1))]),T[226]||(T[226]=m(",满足"))]),Q("mjx-container",d3,[(t(),a("svg",r3,T[227]||(T[227]=[l('',1)]))),T[228]||(T[228]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msub",null,[Q("mi",null,"e"),Q("mn",null,"1")]),Q("msub",null,[Q("mi",null,"d"),Q("mn",null,"1")]),Q("mo",null,"≡"),Q("msub",null,[Q("mi",null,"e"),Q("mn",null,"2")]),Q("msub",null,[Q("mi",null,"d"),Q("mn",null,"2")]),Q("mo",null,"≡"),Q("mn",null,"1"),Q("mspace",{width:"1em"}),Q("mo",{stretchy:"false"},"("),Q("mi",null,"mod"),Q("mspace",{width:"0.333em"}),Q("mi",null,"ϕ"),Q("mo",{stretchy:"false"},"("),Q("mi",null,"n"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))]),T[252]||(T[252]=Q("p",null,"那么",-1)),Q("mjx-container",o3,[(t(),a("svg",H3,T[229]||(T[229]=[l('',1)]))),T[230]||(T[230]=Q("mjx-assistive-mml",{unselectable:"on",display:"block"},[Q("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[Q("msub",null,[Q("mi",null,"E"),Q("mi",null,"B")]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"E"),Q("mi",null,"A")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"M"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")"),Q("mo",null,"="),Q("msub",null,[Q("mi",null,"E"),Q("mi",null,"A")]),Q("mo",{stretchy:"false"},"("),Q("msub",null,[Q("mi",null,"E"),Q("mi",null,"B")]),Q("mo",{stretchy:"false"},"("),Q("mi",null,"M"),Q("mo",{stretchy:"false"},")"),Q("mo",{stretchy:"false"},")")])],-1))])])}const p3=n(s,[["render",i3],["__file","rsa.html.vue"]]),L3=JSON.parse('{"path":"/note/cryptography/rsa.html","title":"RSA算法","lang":"zh-CN","frontmatter":{},"headers":[{"level":2,"title":"1. 历史","slug":"_1-历史","link":"#_1-历史","children":[]},{"level":2,"title":"2. 密钥生成","slug":"_2-密钥生成","link":"#_2-密钥生成","children":[]},{"level":2,"title":"3. 加密和解密","slug":"_3-加密和解密","link":"#_3-加密和解密","children":[]},{"level":2,"title":"4. 举例","slug":"_4-举例","link":"#_4-举例","children":[]},{"level":2,"title":"5. 证明","slug":"_5-证明","link":"#_5-证明","children":[]},{"level":2,"title":"6. 密钥对的个数","slug":"_6-密钥对的个数","link":"#_6-密钥对的个数","children":[]},{"level":2,"title":"7. 交换律","slug":"_7-交换律","link":"#_7-交换律","children":[]}],"git":{},"filePathRelative":"note/cryptography/rsa.md"}');export{p3 as comp,L3 as data};